Question: $\overline{BC} = 8$ $\overline{AC} = {?}$ $A$ $C$ $B$ $?$ $8$ $ \sin( \angle ABC ) = \frac{ \sqrt{5}}{5}, \cos( \angle ABC ) = \frac{2\sqrt{5} }{5}, \tan( \angle ABC ) = \dfrac{1}{2}$
Solution: $\overline{AC}$ is the opposite to $\angle ABC$ $\overline{BC}$ is adjacent to $\angle ABC$ SOH CAH TOA We know the adjacent side and need to solve for the opposite side so we can use the tan function (TOA) $ \tan( \angle ABC ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\overline{AC}}{\overline{BC}}= \frac{\overline{AC}}{8} $ $ \overline{AC}=8 \cdot \tan( \angle ABC ) = 8 \cdot \dfrac{1}{2} = 4$